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Random ergodic theorems with universally representative sequences. (English) Zbl 0813.28004
Let \(\{S_ i: i\in I\}\) be a family of measure preserving transformations acting on a probability space \((Y,{\mathcal C},\nu)\). A scheme for choosing commuting measure preserving transformations at random in a stationary way is provided by a shift invariant measure \(P\) on \(\Omega= I^{\mathbb{N}}\). Put \[ A^ \omega_ n g(y)= {1\over n} \sum^ n_{k= 1} g(S_{\omega_ k} S_{\omega_{k- 1}}\cdots S_{w_ 1} y). \] The authors study conditions which imply that for \(P\)-almost all \(\omega= (\omega_ 1,\omega_ 2,\dots)\) the above averages converge \(\nu\)-almost everywhere for all systems \((Y,{\mathcal C},\nu,\{S_ i\}, g)\) with \(g\in L_ p\) (\(p> 1\) is fixed). In contrast to the classical random ergodic theorems, the exceptional \(P\)-nullset is not allowed to depend on \(g\). For example, if \(I= \{-1,+ 1\}\) and \(S_ i= S^{-1}_{-i}\), the coin tossing measure in \(\Omega\) does not have the desired property. If \(I= \mathbb{Z}\), and \(P\) is a product measure \(P^{\mathbb{N}}_ 1\), and \(S_ i= S^ i\), then the property holds if and only if the expectation of \(P_ 1\) is \(\neq 0\). Choosing among commuting transformations according to the entries of a certain Sturmian sequence does provide a \(P\) having the above property (called universally representative scheme). There are numerous further interesting examples and theorems. The return time theorem of Bourgain, Furstenberg, Katznelson and Ornstein corresponds to the special case when the sampling times take values in a lattice in \(\mathbb{R}\).

MSC:
28D05 Measure-preserving transformations
28D10 One-parameter continuous families of measure-preserving transformations
28D15 General groups of measure-preserving transformations
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